boycotting things was a big part of the civil rights movement in the 50s and 60s!
in 1956 after Rosa Parks refused to give up her seat to a white person, civil rights leaders in Montgomery launcheda boycott of city buses, which MLKjr was a big supporter of.
But the boycotts were specific. they were targeted at businesses and institutions, not at other black citizens or people fighting with the civil rights movements.
The civil rights movements were born in the belly of the beast in the south. They had to do sit-ins, protests, marches, and boycotts in the places that were the most hostile towards them in order to be effective.
turning your back on places where injustices happen, doesn't make them go away and the civil rights movement's concentration on the exact areas where they were not wanted proved to be highly effective.
Fair enough, MLK was into boycotting, just not the whole state